Engineering Rock Mechanics

2 1 6 Rock dynamics and time-dependent aspects

Generalized Kelvin substance Burgers substance

Elastoviscoplastic (Bingham) substance Plastoviscoelastic substance

Figure 13.6 Multi-component rheological models.

of presentation. From the matrix structure, these substances are therefore
termed elastoviscoplastic and plastoviscoelastic materials, respectively. Also
note that the elastoviscoplastic substance’s behaviour is dictated in turn by
the spring, the dashpot and the slider, i.e. by its elastic, viscous and plastic
elements in sequence. Conversely, when the elements are linked in parallel
(the plastoviscoelastic substance), the total model behaviour is dictated in
turn by the slider, dashpot and spring. The off-diagonal rheological
substances above the leading diagonal in Fig. 13.5 are series models,
whereas those below the leading diagonal are parallel models.
With multi-component models there are many ways in which the com-
ponents can be combined in series and parallel sub-networks. In Fig. 13.6
we have shown one extension of the Maxwell model, the combination of
the Maxwell and Kelvin models, and the two simplest ways of combining
the three basic rheological elements. In theory, and by analogy with
electrical resistors, capacitors and fuses, we could generate any n-com-
ponent model and establish its global constitutive behaviour. Such time-
dependency models contain a large number of terms and so are lengthy
and can be difficult to assimilate. It is, therefore, instructive to consider,
mathematically, the simpler Maxwell (elastoviscous) and Kelvin (viscoelas-
tic) models (i.e. elements 2,2 and 2,2 of the matrix in Fig. 13.5) as examples
of time-dependent behaviour.
The two fundamental elements, viscous and elastic, have basic uniaxial
constitutive laws of rs = Fdddt and rs = EE, respectively, where F and E
are the uniaxial constants of viscosity and elasticity.
The Maxwell model consists of viscous and elastic elements in series.
Consequently, the stress is identical in each of the elements and the strain
developed in the material, is the sum of the strains developed in the
elastic and viscous elements, i.e. and cV, respectively. Thus,